On the Classification of Residues of the Grassmannian

TL;DR

This paper develops a new coordinate system for classifying residues of the Grassmannian integral in scattering amplitudes, simplifying the identification of independent residues and solutions.

Contribution

It introduces an iterative formula for Yangian invariants and new Grassmannian coordinates that unify the treatment of composite and non-composite residues.

Findings

New coordinates reveal all residues uniformly.

Residue theorems are formulated in the new variables.

Classification of residues is simplified for example cases.

Abstract

We study leading singularities of scattering amplitudes which are obtained as residues of an integral over a Grassmannian manifold. We recursively do the transformation from twistors to momentum twistors and obtain an iterative formula for Yangian invariants that involves a succession of dualized twistor variables. This turns out to be useful in addressing the problem of classifying the residues of the Grassmannian. The iterative formula leads naturally to new coordinates on the Grassmannian in terms of which both composite and non-composite residues appear on an equal footing. We write down residue theorems in these new variables and classify the independent residues for some simple examples. These variables also explicitly exhibit the distinct solutions one expects to find for a given set of vanishing minors from Schubert calculus.